Some Homology and Cohomology Calculations for the quotient map $q:S^n\rightarrow \mathbb{R}P^n$ I am trying to calculate maps induced by the quotient map $q:S^n\rightarrow \mathbb{R}P^n$, i.e. the descriptions of the maps:
$(1)q_*:H_n(S^n)\rightarrow H_n(\mathbb{R}P^n)$
$(2)q^*:H^n(\mathbb{R}P^n)\rightarrow H^n(S^n)$
$(3)q^*:H^n(\mathbb{R}P^n;\mathbb{Z}/2\mathbb{Z})\rightarrow H^n(S^n;\mathbb{Z}/2\mathbb{Z}) $
$(1)$ is only interesting when $n$ is odd, it feels the map should be multiplication by $2$ in that case, but can't find a detailed proof. 
$(2)$ is zero map.
$(3)$ I can only see that both sides are $\mathbb{Z}/2\mathbb{Z}$
I know cellular homology and cohomology and universal coefficient theorem but don't know cofibration sequence. Any help is appreciated.  
 A: First a reference to read, How to compute the homology of $\mathbb{R}P^n$ with cellular homology:Peter May concise course in algebraic topology page 105. he gives $\mathbb{R}P^n$ a CW structure with one $q$-cell for $0 \le q \le n$ by passage
to quotients from a CW structure on $S^n$ with two $q$-cells for $0 \le q \le n$.
The cells are defined by $$j^q_{\pm}(x_1, \dots, x_q) = (\pm x_1, . . .,±x_q,±(1-\sum
x_i^2)^{1/2}).$$
He then computes the differentials of the cellular chains of $S^n$, to deduces that the differentials are:
$$d_q[j_{+}^q]=(-1)^q d_q[j_{-}^q]=j=[j_+^{q-1}]+(-1)^q[j_{-}^{q-1}]$$
for all $q \ge 1$. We can conclude that a generator for $H^n(S^n)$ is $[j_+^{n}]+(-1)^{q+1}[j_{-}^q]$.  
He then defines the cell structure of $\mathbb{R}P^n$. If $p:S^n \to \mathbb{R}P^n$ is the quotient map, he defines the cellular structure on $\mathbb{R}P^n$ by the cellular maps $j^q=p \circ j_+^q=p \circ j_-^q$. Because he choose the CW structure for $S^n$ wisely (The antipodal map acts nice on that structure), he can easily calculate the differentials for the cellular complex of $\mathbb{R}P^n$ to find:
$$d[j^q] = (1 + (−1)^q)[j^{q−1}] $$
 we can now deduce that the generator $[j_+^{n}]+(-1)^{n+1}[j_{-}^n]$ of $H^n(S^n)$ is sent by $p_*$ to $(1+(-1)^{q+1})j^n$. You can now conclude that $p_*$ is zero for $n$ even and is multiplication by $2$ for $n$ odd.
For your other 2 questions you should dualize your chain complex and do the same kind of calculations. 
